The Life and Work of Srinivasa Ramanujan

The Life and Work of Srinivasa Ramanujan

Srinivasa Ramanujan is considered as one of the mathematical geniuses of all time. He was born near the rural parts of Madras, India.  [2] His remarkable contributions to pure mathematics placed him in the ranks of Gauss, Galois, Abel, Euler, Fermat, Jacobi, Riemann and other similar Mathematician.  His contributions to the theory of numbers are generally considered unique and astonishing. These were even more remarkable by the fact much of his work was done without any formal education in pure mathematics. During his life-time, Ramanujan became a living legend and a versatile creative intellect. His name will be encountered in the history of mathematics as long as humanity will study mathematics.

[3] Ramanujan was born on December 22, 1887 in his grandparent’s house in the town of Erode, a small town in Southern India.  Although the family was very poor, the town’s people respected the family since they belonged to the Brahmin cast, a priestly and a scholarly cast of the Hindu religion.  When Ramanujan was 1-year-old, his father moved the family to Kumbakonam, a nearby town from Erode. [2] His father took a position as a clerk in a cloth-merchant’s office in Kumbakonam, and used to maintain his family with a small income.  [2] The mother was a singer in the local temple and earned some money.  The family also took in boarders in the house that they resided in to earn extra income.  At the age of five, Ramanujan started primary school and finished it by the age of 10.  He was a brilliant student and received distinction of being the best student in the local district.  At the age of 10, he started High school.  In the high school library, he discovered mathematic books which was the start of his journey of mathematics discovery.  [1] One book in particular that that was given to him by a friend that intrigued and influenced him was A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr. [5] The book contained over 4,000 theorems without any proofs, cramming into its pages as many results as possible.

[pic 1]

Entry 2478 from Carr’s Synopsis of elementary results in pure mathematics

Ramanujan was consumed by the book and spent time proving the theorems in the book in detail.  [5] With little other guidance, Ramanujan came to believe this was how mathematics was done, so he himself learned to show little working. Also, he could afford only a small amount of paper, doing most of his work on slate with chalk, transferring a minimal amount of his working to paper. After he completed this book, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler-Mascheroni constant up to 15 decimal places.   

[2] In 1904, Ramanujan Joined the Government College at Kumbakonam with the Subramanian Scholarship which is usually awarded to students for proficiency in Mathematics and English.  He spent most of his time reading and studying mathematics books.  [2] His focus on Mathematics was so much that he neglected all of his other subjects in college.  One example was that he refused to dissect a frog for his physiology class since it was against his Hindu religion and thus he failed that class. The neglect resulted in him failing all his other subjects and resulted in him losing his scholarship.  The lack of being able to pay for school and the disappointment of not progressing, so he dropped out of school.  In the summer 1905, Ramanujan took a month hiatus in a nearby beach town. After which, he enrolled at Pachaiyappa's College in Madras.  He again excelled in mathematics, but he performed poorly in other subjects, such as English, physiology and Sanskrit. Ramanujan failed his fellow of Arts exam in two consecutive years, 1906 and 1907. Without a degree, he left college and continued to pursue independent research in mathematics the next couple of years, living in extreme poverty and often on the brink of starvation.  

[3] At the age of 22 in 1909, Ramanujan married his wife Janakiammal or Janaki for short.  The marriage was an arranged by the parents.  Janaki was only nine years old, so after the marriage, she went back to live with her parents till she was of age to be with her husband.  Since dropping out of college in 1906, Ramanujan was not able to obtain employment, but needed to find a job to support his wife and himself.  In the city of Madras he found some students who needed mathematics tutoring and he also walked around the city offering to do accounting work for businesses.  He applied for a job with Indian Civil Service and by luck the man that had interviewed him, Ramaswamy Ayyar, was the founder of the Indian Mathematical Society.  After Mr. Ayyar started interacting with Ramanujan and reviewing some of his work, he realized that Ramanujan was a special person. Mr. Ayer said this about Ramanujan’s work [1] “I was struck by the extraordinary mathematical results contained in it. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.”  Mr. Ayyar was able to find a job for Ramanujan that would allow him time to do his research. He finally accepted a job in 1912 with the Madras Port Trust as a clerk.  Janaki and her mother joined Ramanujan in 1912 after he secured a job. This was a stable job that allowed him plenty of time to continue his research in mathematics.  He earned 20 rupees per month.

During this period Ramanujan was completely focused on Mathematics. A friend who knew him from college noted that when Ramanujan was concentrating on some mental tasks, the pupils of his eyes disappear. [2] This type of intensity at which he worked resulted in a publication in the Indian Mathematical Society in 1911 and in the next year he published two additional papers.  His former Professor Seshu Aiyer was so impressed with his work after reviewing the publications that he had requested Ramanujan to write a letter to then famous mathematician G.H. Hardy, a fellow of trinity college, Cambridge.  In Ramanujan wrote a letter to Dr. Hardy that was 11 pages long with divergent series without proofs.  [4] The letter stated this to Dr. Hardy:

“I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £ 20 per annum. I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling “I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me “